Variants of the Busemann-Petty problem and of the Shephard problem

摘要

We provide an affirmative answer to a variant of the Busemann-Petty problem,proposed by V.~Milman: Let $K$ be a convex body in ${\mathbb R}^n$ and let $D$be a compact subset of ${\mathbb R}^n$ such that, for some $1\ls k\ls n-1$,$$|P_F(K)|\ls |D\cap F|$$ for all $F\in G_{n,k}$, where $P_F(K)$ is theorthogonal projection of $K$ onto $F$ and $D\cap F$ is the intersection of $D$with $F$. Then, $$|K|\ls |D|.$$ We also provide estimates for the lowerdimensional Busemann-Petty and Shephard problems, and we prove separation inthe original Busemann-Petty problem.